Abstract: The bounds on the discrepancy of approximate solutions constructed by Gedunov's scheme to IVP of isentropic equations of gas dynamics are obtained, Three well-knowu results obtained by Lax for shock waves with small jumps for general quasilinear hyperbolic systems of conservation laws are extended to shock waves for isentropic equations of gas dynamics in a bounded invariant region with ρ=0 as one of boundries of the region. Two counterexamples are given to show that two inequalities given by Godunov do not hold for all rational numbers γ∈(1, 3]. It seems that the approach by Godunov to obtain the forementioned bounds may not be possible.